Higher Green’s functions for modular forms

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Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.) der

Mathematisch-Naturwissenschaftlichen Fakult¨at der

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von Anton Mellit

aus Kiew

Bonn 2008

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1. Gutachter: Prof. Dr. Don Zagier 2. Gutachter: Prof. Dr. G¨unther Harder

Tag der Promotion: 23.06.2008

Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss online elektronisch publiziert.

Erscheinungsjahr: 2009

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Introduction

The subject of the present thesis is higher Green’s functions. For any integerk >1 and subgroup Γ⊂P SL₂(Z) of finite index there is a unique function G^H/Γ_k on the product of the upper half planeHby itself which satisfies the following conditions:

(i) G^H/Γ_k is a smooth function on H×H\ {(τ, γτ)|γ ∈Γ, τ ∈H} with values in R.

(ii) G^H/Γ_k (γ₁τ₁, γ₂τ₂) =G^H/Γ_k (τ₁, τ₂) for allγ₁, γ₂∈Γ.

(iii) ∆_iG^H/Γ_k =k(1−k)G^H/Γ_k , where ∆_i is the hyperbolic Laplacian with respect to the i-th variable, i= 1,2.

(iv) G^H/Γ_k (τ₁, τ₂) = mlog|τ₁−τ₂|²+O(1) when τ₁ tends to τ₂ (m is the order of the stabilizer ofτ₂, which is almost always 1).

(v) G^H/Γ_k (τ₁, τ₂) tends to 0 when τ₁ tends to a cusp.

This function is called the Green function. It is necessarily symmetric, G^H/Γ_k (τ₁, τ₂) =G^H/Γ_k (τ₂, τ₁).

Such functions were introduced in paper [GZ86]. Also it was conjectured in [GZ86] and [GKZ87] that these functions have “algebraic” values at CM points. A particularly simple formulation of the conjecture is in the case when there are no cusp forms of weight 2kfor the group Γ:

Conjecture (1). Suppose there are no cusp forms of weight 2kfor Γ. Then for any two CM pointsτ₁, τ₂ of discriminants D₁,D₂ there is an algebraic number α such that

G^H/Γ_k (τ₁, τ₂) = (D₁D₂)^1−k² logα.

The main result of this thesis is a general approach for proving this conjecture and an actual proof for the case Γ =P SL₂(Z),k= 2,τ₂= i¹. The numberα in the latter case is represented as the intersection number of a certain higher Chow cycle on the elliptic curve corresponding to the pointτ₁ and an ordinary algebraic cycle which has a certain prescribed cohomology class.

The general approach can be formulated as follows. For an elliptic curveE we consider the higher Chow groupCH^k(E^2k−2,1) on the product ofE by itself 2k−2

1We use “i” to denote√

−1 and “i” for other purposes.

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times. The elements of this group are represented by so-called “higher cycles”, which are formal linear combinations X

i

(W_i, f_i),

whereW_i is a subvariety ofE^2k−2 of codimensionk−1 andf_i is a non-zero rational function on W_i such that the following condition holds in the group of cycles of

codimensionk: X

i

divf_i= 0.

Denote the abelian group of higher cycles byZ^k(E^2k−2,1), so thatCH^k(E^2k−2,1) is its quotient (the relations definingCH^k(E^2k−2,1) will be explained in Section 2.3).

We have the Abel-Jacobi map

AJ^k,1 :CH^k(E^2k−2,1)−→ H^2k−2(E^2k−2,C)

F^kH^2k−2(E^2k−2,C) + 2πiH^2k−2(E^2k−2,Z), and there is a canonical cohomology class [θ]∈ H²(E²,C), namely the one represented by the form

θ= ω⊗ω¯+ ¯ω⊗ω R

Eω∧ω¯ ,

where ω is a holomorphic differential 1-form on E. On may notice that for any element [x]∈CH^k(E^2k−2,1) there is a perfectly defined number

<

³

i^k−1(AJ^k,1[x],[θ]^k−1)

´ .

So we hope that for those cases for which the conjecture above is formulated, for a fixed CM pointτ₂, there exists a family of higher cycles{x_s}_s∈S,x_s∈CH^k(E_s^2k−2,1) for a family of elliptic curves {E_s}_s∈S (S is an algebraic variety over C) which

“computes” the values of the Green function in the way described above. This approach appeared from an attempt to understand the discussion of the algebraicity conjecture in [Zha97].

To be more specific I introduce the following 3 classes of functions.

Let A be a subgroup of C (we will usually take A =Z, A= Qor A = i^k−1R).

Consider a holomorphic multi-valued function f on the upper half plane which is allowed to have isolated singularities and is defined up to addition of polynomials of τ with coefficients in 2πiA of degree not greater than 2k−2. This means that for a small disk U which does not contain the singularities off one has an element of O_an(U)/2πiV_2k−2Â , whereV_2k−2Â is the abelian group of polynomials with coefficients in A of degree not greater than 2k−2. Suppose f transforms like a modular form of weight 2−2k with respect to Γ, i.e. f ≡f|_2−2kγ mod 2πiV_2k−2Â for any γ ∈ Γ.

Denote the abelian group of all such functions by M_2−2k^A (Γ). Note that if we put some growth condition on f near the singularities, it will imply that the singularity of f at a pointτ₀ will be of the form F(τ) log(τ −τ₀) for F ∈V_2k−2^A .

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Similarly, let M_0,k^A (Γ) be the space of multi-valued functions f with isolated singularities satisfying ∆f = k(1−k)f, almost holomorphic (expressible as polynomials of _τ−¯¹_τ with holomorphic coefficients), invariant (in weight 0) with respect to Γ, defined up to addition of functions of the form P_2k−2

i=0 α_ir_i(τ), where α_i ∈2πii!(2k−2−i)!

(k−1)! A and ther_i are defined by the equation

2k−2X

i=0

r_i(τ)Xⁱ =

µ(X−τ)(X−τ¯) τ −τ¯

¶_k−1 .

Finally, letM_2k^A(Γ) be the space of (single-valued) holomorphic functionsf with isolated singularities which transform like modular forms of weight 2k with respect to Γ and the V_2k−2^C -valued differential form ω_f = f(τ)(X−τ)^2k−2dτ satisfies the following two conditions:

(i) If we integrate ω_f around a singularity we obtain an element of the abelian group 2πi(2k−2)!V_2k−2^A .

(ii) The cohomology class [ω_f]∈H¹(Γ, VC/2πi(2k−2)!A

2k−2 ) of ω_f is trivial.

Then there is a commutative diagram:

M_2−2k^A (Γ) ^δ^k−1 ^//

(_dτ^dL)L^2k−1LLLLLLLLM%% _0,k^A(Γ)

δ^k

²²

M_2k^A(Γ)

Here the horizontal arrow is the operatorδ₋₂· · ·δ_2−2kand the vertical one isδ_2k−2· · ·δ₀ (δ_w = _dτ^d +_τ−¯^w_τ). Also the horizontal arrow is an isomorphism and the vertical one is surjective with kernel the finite group (V_2k−2/V_2k−2^A )^Γ. In particular ifA isQorR, then the vertical arrow is also an isomorphism. WhenA=Rwe also have a fourth group in our system. Denote by C_0,k^ω (Γ) the space of real-analytic (single-valued) functions with isolated singularities satisfying the same differential equation as for the spaceM_0,k^A (Γ) and invariant (in weight 0) with respect to the action of Γ. Then the following diagram is commutative and all its arrows are isomorphisms:

M_2−2kⁱ^k−1^R(Γ) ^δ^k−1^//

(dτ^d)^2k−1

²²

M_0,kⁱ^k−1^R(Γ)

2<(·)

²²

M_2kⁱ^k−1^R(Γ) C_0,k^ω (Γ)

δ^k

oo

Now we give two ways to construct elements of the spaces above. The first way takes the Green function as input. Fix a point τ₀ ∈ H. The function G^H/Γ_k (τ, τ₀) belongs to the spaceC_0,k^ω (Γ). Therefore our construction produces elements in the spacesM_2−2kⁱ^k−1^R(Γ),M_0,kⁱ^k−1^R(Γ),M_2kⁱ^k−1^R(Γ). Denote by g_τ₀ =g_τ^H/Γ₀_,k the corresponding element inM_2kⁱ^k−1^R(Γ). We prove thatg_τ₀ is meromorphic, zero at the cusps, and its

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principal part atτ₀ism(−1)^k−1(k−1)!Q_τ₀(τ)^−k(we denoteQ_τ₀(X) = ^(X−τ_τ⁰^)(X−¯^τ⁰⁾

0−¯τ0 ).

The integral aroundτ₀ofω_g_τ₀ =g_τ₀(τ)(X−τ)^2k−2dτ is 2πi^(2k−2)!_(k−1)!Q_τ₀(X)^k−1. There- fore the product (k−1)!D₀^k−1² g_τ₀ satisfies all the requirements of M_2k^Z(Γ) except, possibly, the last one. The last requirement is automatically satisfied in the case S_2k(Γ) ={0}, at least up to torsion, i.e. there exists an integer N₁ (depending only on k and Γ) such that N₁D₀^k−1² g_τ₀ ∈ M_2k^Z(Γ). The lift of N₁D₀^k−1² g_τ₀ to M_2−2k^Z (Γ) andM_0,k^Z (Γ) is defined up to a finite group, therefore its product with certain integer numberN₂(which depends only on Γ andk) is perfectly defined. LetN =N₁N₂. We see that the construction canonically gives a function Gb^H/Γ_k,τ₀ ∈ _N¹D₀^1−k² M_0,k^Z (Γ) such thatδ^kGb^H/Γ_k,τ₀ =g_τ₀. If, moreover,τ is a CM point of discriminantD, thenGb^H/Γ_k,τ₀(τ) is defined up to 2πi_N¹(D₀D)^1−k² Z. This is called “the lifted value of the Green function”

and we conjecture that it equals _N¹(D₀D)^1−k² logα for some α ∈ Q^×. We empha- size that the Green function produces elements of spaces M_2−2kⁱ^k−1^R(Γ), M_0,kⁱ^k−1^R(Γ), M_2kⁱ^k−1^R(Γ), and only if τ₀ is a CM point we can “lift” these elements to spaces M_2−2k^Z (Γ),M_0,k^Z (Γ),M_2k^Z(Γ).

There is another way to obtain functions as above, which always produces elements of spaces M_2−2k^Z (Γ), M_0,k^Z (Γ), M_2k^Z(Γ). Suppose we have a family of elliptic curves{E_s}_s∈S over an algebraic varietyS defined overCand an algebraic family of higher cyclesx={x_s ∈Z^k(E_s^2k−2,1)}_s∈S. Letx_s=P

i(W_i,s, f_i,s) (the dimension of W_i,s isk−1 and P

idivf_i,s= 0). Suppose also a map ϕ:S →H/Γ is given which is dominant, i.e. its image is H/Γ without a finite number of points, and suppose that ϕmakes the following diagram commutative (j is the j-invariant):

S _ϕ ^//

jJJJJJ%%

JJ JJ

JJ H/Γ

²²

H/P SL₂(Z).

Definition. A tripleX= ({E_s}_s∈S,{x_s}_s∈S, ϕ) as above is calledmodular if for any two pointss₁, s₂ such thatϕ(s₁) =ϕ(s₂) there exists an isomorphismρ:E_s₁ →E_s₂ with the following properties:

(i) ρ^2k−2(W_i,s₁) =W_i,s₂,

(ii) ρ^2k−2(f_i,s₁) =β_if_i,s₂ for β_i∈C^×.

If we have a modular triple then we construct an element ofM_2−2k^Z (Γ) as follows.

Letτ ∈Hbe such that its projection toH/Γ belongs to the image ofϕ, sayτ =ϕ(s).

Choose a differential formωon E_sand suppose its period lattice is generated by Ω₁ and Ω₂ with ^Ω_Ω²

1 =τ. Then put A_X(τ) = 1

Ω^2k−2₁ hAJ^k,1[x_s],[ω^⊗2k−2]i.

In this case the Abel-Jacobi map reduces to the following integral:

A_X(τ) = 2πi 1 Ω^2k−2₁

Z

ξ

ω^⊗2k−2,

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where ξ is a smooth 2k−2-chain whose boundary is P

if_i,s^∗ [0,∞] and [0,∞] is a path from 0 to∞ on CP¹. It is clear that A_X(τ) thus defined does not depend on the choice ofswithϕ(s) =τ, and A_X ∈M_2−2k^Z (Γ). We will show that

δ^k−1A_X(τ) = (2k−2)!

(k−1)!³R

Esω∧ω

´_k−1 hAJ^k,1[x],[ω^⊗k−1^sym⊗ ω^⊗k−1]i.

If τ is a CM point with minimal equation aτ² +bτ +c = 0, D = b² −4ac and ϕ(s) =τ, then there are endomorphisms ofE_swhich act on the tangent space as the multiplications byaτ anda¯τ. Denote their graphs byY_aτ andY_a¯_τ correspondingly.

Then one can check that the Poincar´e dual cohomology class of the differenceY_aτ− Y_a¯_τ is represented by the form−2√

D ^ω

sym⊗ω R

Esω∧ω. Therefore we can construct a variety whose Poincar´e dual class is represented by

(−1)^k−1D^k−1² (2k−2)!

(k−1)!³R

Esω∧ω

´_k−1 ω^⊗k−1^sym⊗ ω^⊗k−1,

namely, we take the product ofk−1 copies of Y_aτ −Y_a¯_τ for each possible splitting of the product E_s^2k−2 into pairs and add them up. Note that there are precisely

(2k−2)!

(k−1)!2^k−1 such splittings. Denote the variety obtained in this way byZ_s. Then we obtain

δ^k−1A_X(τ) = (−1)^k−1D^1−k² log (x_s·Z_s), where the intersection numberx_s·Z_s is defined as

x_s·Z_s = Y

i

Y

p∈Wi,s∩Zs

f_i,s(p)^ord^p^W^i,s^·Z^s

ifZ_s does not intersect the divisors off_i,s, singularities of W_i,s and intersects W_i,s properly. If this is not the case one can still define the intersection number, for example, by “shifting”Z_s using the the addition law.

The second construction gives examples of functions inM_0,k^Z whose values at CM points are “algebraic”. We prove that for such A_X, obtained via the second construction, the derivative ¡_d

dτ

¢_2k−1

A_X(τ) is meromorphic. In fact we give a method to compute this derivative purely algebraically.

Definition. A meromorphic modular formfwhich belongs toM_0,k^Z is calledgeomet- rically representableiff =¡_d

dτ

¢_2k−1

A_X(τ) for some modular triple ({E_s}_s∈S,{x_s}_s∈S, ϕ) with{E_s},ϕand {x_s} are defined overQ.

Then we have

Lemma. Suppose Γ is a congruence subgroup, k >1 and f ∈M_0,k^Z (Γ) is such that (−1)^k−1N₁δ^kf, for some integer N₁, is geometrically representable by a modular triple ({E_s}_s∈S,{x_s}_s∈S, ϕ). Suppose τ ∈ H is a CM point which belongs to the image of ϕ. Let D be the discriminant of τ. Then D^k−1² f(τ) ≡ _N¹

1 logα mod ^2πi_N , whereα is an algebraic number, which can be computed as the intersectionx_s·Z_s as above withϕ(s) =τ. HereN₂ is the exponent of the groupH⁰(Γ, V_2k−2^Q/Z),N =N₁N₂.

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In particular, if a multiple of D₀^k−1² g_k,τ₀ is geometrically representable, then the conjecture is true forτ₁ =τ₀and τ₂ — any other CM point except, possibly, a finite number of points (those which do not belong to the image ofϕ). We give an example when this occurs. Let Γ = P SL₂(Z), k = 2. Let S be the open subset of C×C of pairs (a, b) satisfying 4a³+ 27b² 6= 0 and b6= 0. Let {E_s}_s∈S be the Weierstrass family, which is defined by the projective version of the equation y² =x³+ax+b.

LetW_sbe the subvariety ofE_s×E_swhich consists of (x₁, y₁, x₂, y₂) withx₁+x₂= 0.

Let f_s be the function y₁−iy₂. It is easy to check that (W_s, f_s) ∈Z²(E_s×E_s,1).

Let ϕ be the j-invariant. One can verify that this gives a modular triple. Denote it by X₋₄ (since the discriminant of the point τ = i, which is the only point which does not belong to the image ofϕ, is −4). We prove

Theorem. Consider the following modular form of weight 4:

−1 2

√−4δ²G^{H/P SL}₂ ²^(Z)(τ,i) = (2πi)²√

−4 432E₄(τ) j(τ)−1728. This form is geometrically representable by the modular triple X₋₄.

This implies the conjecture for G^{H/P SL}₂ ²^(Z)(τ,i) (keep in mind that G= 2<G),b namely

Corollary. For any CM pointτ which is not equivalent to i one has

√−4DGb^{H/P SL}₂ ²^(Z)(τ,i)≡2 log ((W_s, f_s)·(Y_Aτ −Y_A¯_τ)) mod πiZ,

where s= (a, b), the curve y² =x³+ax+b corresponds to τ, Y_Aτ and Y_A¯_τ are the graphs of the endomorphisms of this curve which act on the tangent space asAτ and A¯τ, Aτ²+Bτ+C is the minimal equation ofτ with A >0, and D=B²−4AC.

As an example we verify Corollary.

G^{H/P SL}₂ ²^(Z)

µ−1 +√

−7 2 ,i

¶

= 8

√7log(8−3√ 7).

The text is organized in five chapters. Each chapter is provided with a more detailed introduction and reading the introduction is highly recommended for un- derstanding the chapter, especially in the case of Chapter 3. We briefly discuss contents of each chapter here. The first chapter studies various functions on the upper half plane and differential operators. The main result of this chapter is the lifting of the values of the Green function at CM points from real numbers to the elements of

C/2πi(D₁D₂)^1−k² Q,

and, related to this, the refined version of the algebraicity conjecture (see Corol- lary 1.5.6 and Conjecture (2)). Also in this chapter we prove that the k-th non- holomorphic derivative of G^H/Γ_k is a meromorphic modular form characterized by

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certain properties (Theorem 1.5.3) and discuss ways to compute the Green function from this modular form (see the discussion in the end of Section 1.5). The results of this chapter where known to experts but do not exist in print.

The second chapter contains a definition of the higher Chow groups and a construction of the Abel-Jacobi map as in [GL99] and [GL00], as well as a proof that the definition of the Abel-Jacobi map is correct. We prove a formula (Theorem 2.5.1) which in certain cases relates the value of the Abel-Jacobi map with certain intersection number which takes values in the multiplicative group.

The third chapter provides a way to compute the derivative of the Abel-Jacobi map (by this we mean¡_d

dτ

¢_2k−1

A_X(τ)) for a family of higher cycles. The answer is expressed as a certain extension ofD-modules. The result is more general than we need and works for any families of products of curves. In the case of the power of a family of elliptic curves we obtain an invariant of this extension which for modular triples coincides with¡_d

dτ

¢_2k−1

A_X(τ) and is a meromorphic modular form.

In the fourth chapter we study various objects on the Weierstrass family of elliptic curves. We provide representatives for cohomology classes which are necessary for later computations.

The fifth chapter contains the main result, namely, the construction of a family of higher cycles which computes the values of the functionG^H/Γ₂ (τ,i) and the proof of the algebraicity conjecture in this case (Theorem 5.2.1). There we show that the meromorphic modular form (2πi)²√

−4_j(τ)−1728^432E⁴^(τ), which equals−¹₂√

−4δ²G^H/Γ₂ (τ,i), is geometrically representable.

The construction of the family of higher cycles used in the proof was inspired by other constructions of higher cycles on products of elliptic curves in [GL99], [GL00].

It seems, however, that our family (see Section 5.1.3 and the construction of X₋₄ above) was not known before, though its construction is surprisingly simple.

This text is submitted as a PhD thesis to the University of Bonn. The work was done at the Max-Planck-Institute for Mathematics. The author is grateful to the institute for hospitality and good working atmosphere. Also I wish to thank S.

Bloch, J. Bruinier, N. Durov, A. Goncharov, G. Harder, D. Huybrechts, C. Kaiser, Yu. Manin, R. Sreekantan with whom I had interesting discussions on the subject of this thesis. Special thanks to M. Vlasenko who read drafts of this text and made useful remarks, and to D. Zagier who introduced me to number theory, proposed the problem, provided me with inspiration and support, and was a very good supervisor.

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Modular forms

In this chapter we first fix some notation for the representations of SL₂(R) on the space of functions on the upper half plane. We obtain a representation for each integerwwhich is called “weight”. We introduce three differential operators on these representations which are intertwining: the non-holomorphic derivative δ or the

“raising operator”, the “lowering operator”δ⁻and the Laplacian ∆. We also recall the construction of the standard finite-dimensional representations V_m of SL₂(R) and vectors in these representations which correspond to complex multiplication (CM) points. This is done in Section 1.1.

Next, in Section 1.2, we study in more detail functions which are eigenfunctions for the Laplacian with eigenvalue of the form k(1−k), k∈ Z. The situation with these particular eigenvalues is special. In particular, to each eigenfunction f with such eigenvalue corresponds a certain “extended function”fewhich is harmonic, but takes values in the space V_2k−2. The derivatives of these extended functions are proportional to holomorphic functions of weight 2kwhich we also call “derivatives”.

In Section 1.3 we study the inverse problem of recovering fefrom its derivatives in the case when they are invariant under the action of a congruence subgroup of P SL₂(Z). We obtain that the obstructions to solving this problem lie in certain cohomology groups.

Then, in Section 1.4, we define the Green functionsG^H_k for the upper half plane (without a congruence subgroup), the so-called “local Green functions”. These are defined as functions of two variablesz₁,z₂in the upper half plane which areSL₂(R)- invariant (diagonal action) eigenfunctions for the Laplacian with eigenvalue k(1− k) and have only a logarithmic singularity on the diagonal. We show how these functions can be explicitly written using Legendre’s functions. We also evaluate the actions of powers of the operatorδon them. We obtain particularly nice expressions for thek-th power which also gives formulae for the derivatives of the corresponding extended Green functions.

Section 1.5 studies the “global Green functions”G^H/Γ_k for a quotient of the upper half plane. Since these Green functions can be obtained by averaging the local ones, results of Section 1.4 give us information about the singularities of the global Green functions and their derivatives. We obtain a characterisation of the derivativeg^H/Γ_k,z

0 of the extended global Green function as a meromorphic modular form with described

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type of poles and having trivial class in a certain Eichler-Shimura cohomology group.

Then the original Green function can be obtained from the modular form using the integration procedure described in Section 1.3.

We suppose that there are no cusp forms of weight 2k for the group Γ. Then being applied to CM points the integration procedure allows us to lift the value of the Green function, which is real, to a complex number defined up to an algebraic multiple of 2πi (see Corollary 1.5.6). This supports and refines the algebraicity conjecture formulated in papers [GZ86] and [GKZ87], which says that in this case the value must be an algebraic multiple of the logarithm of an algebraic number (see Conjectures (1) and (2)). In the end of the section we discuss ways to compute the lifted Green function, and we show that it is a period (Theorem 1.5.7).

1.1 Notations

We will consider the groupSL₂(R). Elements of this group will be usually denoted byγ, and matrix elements bya,b,c,d:

γ =

µa b c d

¶

∈SL₂(R).

This groups acts on the upper half planeH.

1.1.1 Representations of SL₂(R)

The groupSL₂(R) naturally acts on the following linear spaces:

• V =C² — the space of column vectors of length 2,

• V_m — the symmetric m-th power of V.

We explain the way we view elements of V_m. Let e₁, e₂ be the natural basis on V. Then V_m is the space of homogeneous polynomials ine₁,e₂ of degreem. If we substitute e¹ by a new variable X and e² by 1 we obtain a non-homogeneous polynomial in one variable of degree less or equal m. We represent elements ofV_m as polynomials in the variableX of degree less or equalm, i.e.

p∈V_m, p(X) = p₀+p₁X+· · ·+p_mX^m. The group acts onV_m on the right by

(pγ)(X) = (p|_−mγ)(X) = p(γX)(cX+d)^m, γ =

µa b c d

¶

∈SL₂(R).

There is the corresponding action on the left

(γp)(X) = (pγ⁻¹)(X) = p(γ⁻¹X)(−cX+a)^m.

Now consider the dual space to V_m, V_m^∗. Any v ∈ V_m^∗ is a functional on V_m. Suppose its value onpis

v₀p₀+v₁p₁+. . . v_mp_m,

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then we representvas a raw vector (v₀, v₁, v₂, . . . , v_m). For any two numbersx, y∈C we form a vector

v_x,y = (y^m, y^m−1x, y^m−2x², . . . , x^m)∈V_m. Then for any p∈V_m

(v_x,y, p) = p µx

y

¶

y^m =:p(x, y), hence

γv_x,y = v_γ(x,y) = vax+by,cx+dy.

In this way the action of SL₂(R) is given on a subset ofV_m^∗ which spans V_m^∗. The isomorphism betweenV_mandV_m^∗ can be given as follows. Define an invariant pairing between elements of the formv_x,y ∈V_m:

(v_x,y, v_x⁰_,y⁰) = (xy⁰−x⁰y)^m = Xm

i=0

(−1)ⁱ µm

i

¶

x^m−iyⁱx⁰ⁱy^0m−i.

Since it is a homogeneous polynomial of degree m in each pair of variables, this induces an equivariant linear map from V_m^∗ toV_m

(v₀, v₁, . . . , v_m)7→

Xm

i=0

(−1)ⁱ µm

i

¶

v_m−iXⁱ.

This is an isomorphism of representations. It induces invariant pairings onV_m Ã _m

X

i=0

p_iXⁱ, Xm

i=0

p⁰_iXⁱ

!

= Xm

i=0

(−1)ⁱp_ip⁰_m−i

¡_m

i

¢

and on V_m^∗

((v_i),(v_i⁰)) = Xm

i=0

(−1)ⁱ µm

i

¶

v_iv_m−i⁰ .

The pairings above are symmetric formeven and antisymmetric formodd. The following identities hold:

((z−X)^m, p) = p(z) = (p,(X−z)^m) (p∈V_m, z∈C).

1.1.2 Differential operators

LetSbe a discrete subset of the upper half planeHandf(z) be a function onH−S with values inCand w∈Z. Define differential operators

δ_wf = ∂f

∂z + w z−z¯f, δ_w⁻f = (z−z)¯²∂f

∂z¯,

∆_wf = (z−z)¯² ∂

∂z

∂

∂¯zf + w(z−z)¯ ∂

∂¯z.

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We think about w as the weight, attached to the function f. The weight will be always clear from context, so we will omit the subscriptw. We will follow the following agreement: the operatorδ increases weight by 2, the operatorδ⁻decreases weight by 2 and the operator ∆ leaves weight untouched. Taking into account this agreement the following identities can be proved:

δ⁻δ − δ δ⁻ = w, δ δ⁻ = ∆,

δ⁻δ = ∆ + w.

Let the group SL₂(R) act on functions of weight w by the usual formula:

(f|_wγ)(z) = f(γz)(cz+d)^−w.

This is a right action. We also define the corresponding left action (γf)(z) = (f|_wγ⁻¹)(z) = f(γ⁻¹z)(−cz+a)^−w.

Note that this action commutes with the operators δ, δ⁻, ∆, it maps functions defined onH−S to functions defined on H−γS.

It is also convenient to modify the complex conjugation for functions with weight to make it commuting with the action of the group. Forf of weight wwe put

f^∗(z) = (z−z)¯ ^wf(z).

Assign tof^∗ weight −w. We can check that f^∗∗ = (−1)^wf, δ(f^∗) = (δ⁻f)^∗, δ⁻(f^∗) = (δf)^∗,

∆(f^∗) = ((∆ +w)f)^∗, γ(f^∗) = (γf)^∗.

We remark that for the weight 0 the operator∗ is the usual complex conjugation, the operatorδ is the usual _∂z^∂ , and the operator ∆ is the usual Laplace operator for the hyperbolic metric−y²(_∂x^∂²2 +_∂y^∂²2).

We list several formulae, which are convenient to use in computations. We assume that the constant function 1 has weight 0. Consider the functions

X−z, X−z¯ z−¯z,

which are thought as functions inzwith values inV of weights−1 and 1 respectively.

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Then

δ1 = δ⁻1 = 0, (X−z)^∗ = X−z¯

z−z¯,

µX−z¯ z−z¯

¶_∗

= −(X−z), δ(X−z) = −X−z¯

z−z¯, δX−z¯ z−z¯ = 0, δ⁻(X−z) = 0, δ⁻X−z¯

z−z¯ = X−z, δ

Ã

(X−z)^a

µX−z¯ z−z¯

¶_b!

= −a(X−z)^a−1

µX−z¯ z−z¯

¶_b+1 , δ⁻

Ã

(X−z)^a

µX−z¯ z−z¯

¶_b!

= b(X−z)^a+1

µX−z¯ z−z¯

¶_b−1 ,

∆ Ã

(X−z)^a

µX−z¯ z−z¯

¶_b!

= −b(a+ 1)(X−z)^a

µX−z¯ z−z¯

¶_b .

1.1.3 CM points and quadratic forms We will frequently use the following notation:

Q_z = (X−z)(X−z)¯ z−z¯ .

As a function of z Q_z is a function with values inV₂ of weight 0.

A CM point is a point z ∈ H which satisfies a quadratic equation of degree 2 with integer coefficients. Letz be a CM point. Write the minimal equation forz:

az²+bz+c = 0, a, b, c∈Z, a >0.

The discriminant of the minimal equation D_z =b²−4ac is called the discriminant of z. An elementary computation gives

Q_z = aX²+bX+c

√D .

It is clear that a point z ∈H is a CM point if and only if Q_z is proportional to a polynomial with integer coefficients.

1.2 Eigenfunctions of the Laplacian

We consider functions on H−S for discrete subsets S ⊂ H. For integers k, w we denote by F_k,w the space of functions of weight wsatisfying

∆f = (k(1−k) +^w(w−2)₄ )f.

It is easy to check the following properties of the spaces F_k,w:

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Proposition 1.2.1. (i) The spaceF_k,w is invariant under the action of the group SL₂(R) (meaning, of course, that γ∈SL₂(R) changes S to γS).

(ii) The operator∗ maps F_k,w to F_k,−w.

(iii) The operatorδmapsF_k,wtoF_k,w+2. It is invertible for all values ofw, except, possibly, 2k−2 and−2k with the inverse given by

δ_w⁻¹ = 4

(w+ 2k)(w−2k+ 2)δ⁻_w+2.

(iv) The operator δ⁻ maps F_k,w to F_k,w−2. It is invertible for all values of w, except, possibly, 2k and 2−2kwith the inverse given by

(δ⁻_w)⁻¹ = 4

(w−2k)(w+ 2k−2)δ_w−2.

We will occasionally use negative powers of δ and δ⁻ when they can be defined using this proposition.

Next we state some basic facts

Proposition 1.2.2. For integer numbersk, l we have (X−z)^k−l−1

µX−z¯ z−z¯

¶_k+l−1

∈F_k,2l⊗V_2k−2. Proof. Use the formulae listed in the end of Section 1.1.2.

Proposition 1.2.3. Forf, g in F_k,0 and 1−k≤l≤k−1 we have δ^−lf δ^lg = (δ⁻)^lf(δ⁻)^−lg.

Proof. Note that

(δ⁻)^lf = (−1)^l(k+l−1)!

(k−l−1)! δ^−lf, and analogously

δ^lg = (−1)^l(k+l−1)!

(k−l−1)! (δ⁻)^−lg, which implies the required identity.

Let us introduce the following operation. For two functions f, g from F_k,0 we put

f∗g =

k−1X

l=1−k

δ^−lf δ^lg,

which has weight 0. Note that the previous proposition implies f ∗g =

Xk−1

l=1−k

(−1)^l (δ⁻)^−lf (δ⁻)^lg,

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so

f ∗g = f∗g.

It is also easy to see that

∂

∂z(f ∗g) = (−1)^k−1(δ^1−kf δ^kg + δ^kf δ^1−kg).

Consider the function Q^k−1_z . By Proposition 1.2.2

Q^k−1_z ∈F_k,0⊗V_2k−2 as a function ofz.

One can compute that for 1−k≤l≤k−1 δ^lQ^k−1_z = (−1)^l(k−1)!(X−z)^k−1−l

(k−l−1)!

µX−z¯ z−z¯

¶_k−1+l

(1−k≤l≤k−1), δ^kQ^k−1_z = 0.

For f ∈F_k,0 we denote fe = (−1)^k−1

µ2k−2 k−1

¶

f ∗Q^k−1_z

= (−1)^k−1(2k−2)!

(k−1)!

k−1X

l=1−k

(X−z)^k+l−1 (k+l−1)!

µX−z¯ z−z¯

¶_k−l−1 δ^lf.

It is easy to check that

γff = γfe forγ ∈SL₂(R),

where γ acts on feby the simultaneous action on z in weight 0 and X in weight 2−2k. Also

fe = (−1)^k−1ef .

One can compute the scalar product of δⁱQ^k−1_z and δ^jQ^k−1_z as follows:

Lemma 1.2.4.

(δⁱQ^k−1_z , δ^jQ^k−1_z ) =

(0, if i6=−j (−1)^k−1−i¡_2k−2

k−1

¢₋₁

if i=−j.

Proof. We prove by induction on istarting from 1−k. Since δ^1−kQ^k−1_z = (−1)^k−1 (k−1)!

(2k−2)!(X−z)^2k−2, for any polynomial p∈V_2k−2 we have

(δ^1−kQ^k−1_z , p) = (−1)^k−1 (k−1)!

(2k−2)!p(z).

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Hence (δ^1−kQ^k−1_z , δ^jQ^k−1_z ) is not zero only for j=k−1 and in this case (δ^1−kQ^k−1_z , δ^k−1Q^k−1_z ) =

µ2k−2 k−1

¶₋₁ .

If the statement is true for ithen for any j, taking into account that the weight of (δⁱQ^k−1_z , δ^jQ^k−1_z ) is 2i+ 2j,

0 = δ(δⁱQ^k−1_z , δ^jQ^k−1_z ) = (δⁱ⁺¹Q^k−1_z , δ^jQ^k−1_z ) + (δⁱQ^k−1_z , δ^j+1Q^k−1_z ).

Hence

(δⁱ⁺¹Q^k−1_z , δ^jQ^k−1_z ) = −(δⁱQ^k−1_z , δ^j+1Q^k−1_z ).

We see that ifi+j 6=−1, this is zero. Ifi+j=−1, this equals exactly

−(−1)^k−1−i

µ2k−2 k−1

¶₋₁ .

Therefore the original function f can be recovered fromfeas f = (f , Qe ^k−1_z ).

Note also that ∆fe= 0. This is true because

δ⁻δ^kf = (∆ + 2k−2)δ^k−1f = 0.

Let us summarize.

Theorem 1.2.5. Let f ∈F_k,0 fork≥1. Then the function fesatisfies the following properties (note thatF_0,0 is the space of harmonic functions):

fe ∈ F_0,0⊗V_2k−2, f = (f , Qe ^k−1_z ), δ^lf = (f , δe ^lQ^k−1_z ),

∂fe

∂z = (X−z)^2k−2(−1)^k−1δ^kf (k−1)! ,

∂fe

∂¯z = (X−z)¯ ^2k−2 ¯δ^kf

(k−1)! (¯δg:=δ¯g).

In the opposite way, if g∈ F_0,0⊗V_2k−2 is such that ^∂g_∂z is divisible by (X−z)^2k−2 and ^∂g_∂¯_z is divisible by (X−z)¯ ^2k−2, then the function f(z) = (g(z), Q^k−1_z ) satisfies f ∈F_k,0 and g=fe.

Proof. It is only not clear how to prove the “opposite way” part of the statement.

Supposeg satisfies the conditions above. Consider the function g₀ = (g, δ^k−1Q^k−1_z ).

Higher Green’s functions for modular forms

Dissertation

zur

Erlangung des Doktorgrades (Dr. rer. nat.) der

Mathematisch-Naturwissenschaftlichen Fakult¨at der

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von Anton Mellit

aus Kiew

Bonn 2008

Contents

Introduction

Modular forms

1.1 Notations

1.2 Eigenfunctions of the Laplacian